## A forward-backward SDE from the 2D nonlinear stochastic heat equation

We consider a nonlinear stochastic heat equation in spatial dimension $d = 2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $ε > 0$ but divided by a factor of $\sqrt{\log ε^{ −1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the “weak noise” regime. We show that, as $ε ↓ 0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras.

### Duke Scholars

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- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics

### Citation

*Annals of Probability*,

*50*(3), 1204–1253. https://doi.org/10.1214/21-AOP1563

*Annals of Probability*50, no. 3 (May 1, 2022): 1204–53. https://doi.org/10.1214/21-AOP1563.

*Annals of Probability*, vol. 50, no. 3, May 2022, pp. 1204–53.

*Manual*, doi:10.1214/21-AOP1563.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics